Stability BIBO stability: Def: A system is BIBO-stable if any bounded input produces bounded output. otherwise it’s not BIBO-stable.

Asymptotically Stable A system is asymptotically stable if for any arbitrary initial conditions, all variables in the system converge to 0 as t→∞ when input=0. Thm: If a system is A.S. then it is BIBO-stable But BIBO-stable does not guarantee A.S.in general If there is no pole/zero cancellation, BIBO stable  Asymp Stable

Characteristic polynomials Three types of models: Assume no p/z cancellation Sys characteristic polynomial is:

A polynomial is said to be Hurwitz or stable if all of its roots are in O.L.H.P A system is stable if its char. polynomial is Hurwitz A nxn matrix is called Hurwitz or stable if its char. poly det(sI-A) is Hurwitz, or if all eigenvalues have real parts<0

Routh-Hurwitz Method From now on, when we say stability we mean A.S. / M.S. or unstable. We assume no pole/zero cancellation, A.S. BIBO stable M.S./unstable not BIBO stable Since stability is determined by denominator, so just work with d(s)

Routh Table

Repeat the process until s0 row Stability criterion: d(s) is A.S. iff 1st col have same sign the # of sign changes in 1st col = # of roots in right half plane Note: if highest coeff in d(s) is 1, A.S. 1st col >0 If all roots of d(s) are <0, d(s) is Hurwitz

Example: ←has roots:3,2,-1

(1x3-2x5)/1=-7 (1x10-2x0)/1=10 (-7x5-1x10)/-7

Remember this

Remember this

e.g.

Routh Criteria Regular case: (1) A.S. 1st col. all same sign (2)#sign changes in 1st col. =#roots with Re(.)>0 Special case 1: one whole row=0 Solution: 1) use prev. row to form aux. eq. A(s)=0 2) get 3) use coeff of in 0-row 4) continue

Example ←whole row=0

Replace by e

Useful case: parameter in d(s) How to use: 1) form table as usual 2) set 1st col. >0 3) solve for parameter range for A.S. 2’) set one in 1st col=0 3’) solve for parameter that leads to M.S. or leads to sustained oscillation

Example + s+3 s(s+2)(s+1) Kp

Q: find region of stability in K-a plane.